Abstract
This paper studies the solvability of the functional equationg(x+θ)=φ(x)g(x), given an irrationalθ and a step functionf mappingR/Z (with Lebesgue measure) to the unit circle. Results are applied to find parameterized families of representations of non-regular semi-direct product groups and to display irregularities in the uniform distribution of the sequenceZθ.
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